Finite Group DatabasebetaFinite Group Databasebeta| order | 32340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D147⋊AGL(1,𝔽11) |
| unambiguous semidirect product | (C3⋊AGL(1,𝔽11))⋊D49 |
| unambiguous semidirect product | (C49⋊AGL(1,𝔽11))⋊D3 |
| unambiguous semidirect product | (D33⋊D49)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C11×C49 | C22×C5 |
| Frattini | C7 | D21⋊AGL(1,𝔽11) |
| socle | C3×C7×C11 | C2×C5×D7 |
| LCS | G ▷ C3×C11×C49 | |
| UCS | C1 | |
120 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 1617 |
A minimal faithful permutation representation — 2 generators on 63 points:
(1,2,7,3,9)(4,11,10,5,8)(12,48)(13,40)(14,37)(15,56)(16,47)(17,28)(18,39)(19,25)(20,46)(21,60)(22,55)(23,41)(24,50)(26,27)(29,35)(30,54)(31,58)(32,59)(33,51)(34,57)(36,38)(42,44)(43,52)(45,49)(62,63);(2,3)(4,6)(5,8)(7,11)(9,10)(13,15)(14,18)(16,21)(17,24)(19,27)(20,26)(22,31)(23,34)(25,36)(28,40)(29,39)(30,38)(32,33)(35,45)(37,47)(41,48)(42,50)(43,49)(44,53)(46,55)(51,56)(52,57)(54,59)(58,60)(61,62,63)Polycyclic presentation:
2^2,5,3,7,11,7|e^7=g^6,e^a=e^6g,f^a=f^10,g^a=g^6,d^b=d^2,e^b=e^6g,g^b=g^6,f^c=f^4465 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32340, 47);